I haven't really been able to find an answer to this question. I've only noticed that most instruments during the baroque period were tuned a semitone lower (A=415 Hz I believe). So shouldn't it make sense to play baroque pieces in a key which is a semitone lower than the original or tune the instrument down by a semitone? (Particularly interested in pieces written by Bach)
It would make sense to play them on authentic instruments, at the original pitch, and many performance groups make a speciality of doing this. But there's more to authenticity than just the pitch. I don't see a lot of point in tuning modern instruments down. Maybe if voices are involved.
There's also the question of styles of performance. Even within the era of recorded music, there have been changes. We can now compare styles of performing the same 'classical' pieces 100 years ago and today. And it's very instructive to do so! I think we can safely assume there were similar changes in the preceding 100 years, and the 100 before that. And, although we have been left clues, a lot of our ideas of what was 'authentic' are supposition.
It really depends on the instruments you have available. Bach is not an equal temperament composer, and the instruments of his period are not equal temperament instruments. Even more modern versions of flutes, lutes, string instruments, brass and so on have dominant scales and are built for a particular tuning. Mapping modern instruments to a different key or even "merely" retuning will more likely than not change more of the character than the different frequency as such does.
Singers are a different matter: in fact they are usually built quite larger than in baroque times and expedients like boy sopranos have fallen out of use (let alone baroque castrati). So if you have an orchestra with period instruments in period tuning, tuning down the singers is usually considered the least of problems since more often than not there is a dearth of sopranos and particularly tenors anyway, and few singers have absolute pitch.